Local parameter selection in the $C^0$ interior penalty method for the biharmonic equation

TL;DR

This paper proposes an automatic, geometry-based local parameter selection method for the symmetric $C^0$ interior penalty approach to the biharmonic equation, enhancing stability and efficiency across various mesh types and polynomial degrees.

Contribution

It introduces a novel local stability parameter selection strategy that guarantees discrete ellipticity and can be generalized to higher dimensions and different discretizations.

Findings

The method ensures stable discretization with guaranteed ellipticity.

Numerical experiments confirm the reliability and efficiency of the local parameter choice.

The approach is adaptable to various mesh types and polynomial degrees.

Abstract

The symmetric $C^0$ interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic local selection of the involved stability parameter in terms of the geometry of the underlying triangulation for arbitrary polynomial degrees. The proposed choice ensures a stable discretization with guaranteed discrete ellipticity constant. Numerical evidence for uniform and adaptive mesh-refinement and various polynomial degrees supports the reliability and efficiency of the local parameter selection and recommends this in practice. The approach is documented in 2D for triangles, but the methodology behind can be generalized to higher dimensions, to non-uniform polynomial degrees, and to rectangular discretizations. Two appendices present the realization of our proposed parameter selection in various established finite element software packages as well as a detailed documentation of a self-contained MATLAB program for the lowest-order $C^0$ interior penalty method.